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Points On A Circle II (Posted on 2010-07-13) Difficulty: 3 of 5
Refer to Points On A Circle.

(A) Seven points are placed on the circumference of a circle such that the distance between any two of the points, measured along the circumference, is an integer.

What is the smallest radius of the circle, given that each of the distances is unique?

(B) Seven points are placed on the circumference of a circle such that the distance between any two of the points, measured as a straight line, is an integer.

Determine the smallest radius of the circle. What is the smallest radius of the circle, given that it is rational?

Note: In Part (B) each of the distances may or may not be unique.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
A better upper bound (part A) | Comment 2 of 12 |
One set that seems to work is distances between adjacent points of 1,2,4,8,16,10,22 (in that order).  This gives a total circumference of 63, so the circumference is between 42 (see previous post for this lower bound) and 63.

The 21 pairwise distances for the above arrangement are 

1,2,3,4,6,7,8,10,12,14,15,16,22,23,24,25,26,28,29,30,31

I am not claiming that this is optimal.

Edited on July 13, 2010, 1:48 pm
  Posted by Steve Herman on 2010-07-13 13:39:20

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